I am trying to develop intuition for why a flat map of schemes $f : X \rightarrow Y$ has "continuously varying fibers", and vice versa. Evidently defining what it means for fibers to vary continuously is quite hard, but I am content with working through examples to get a sense of what it means.
Here are my questions:
1) Let $f : M \rightarrow N$ be a map of smooth manifolds. Suppose $C^{\infty}(N ) \rightarrow C^{\infty}(M)$ is flat. Is $f$ open?
2) Let $f: X \rightarrow Y$ be a flat map of schemes. Is $f$ an open map?
3) Let $M$ be $1$ dimensional $\{ (t, t^2) : t \in \mathbb{R} \}$ in $\mathbb{R}^2$. Define a map $M \rightarrow \mathbb{R}$ sending $(t, t^2)$ to $t^2$. Is $C^{\infty}(\mathbb{R}) \rightarrow C^{\infty}(M)$ flat?
4) Let $X = \text{Spec}(k[x, y]/ \langle x^2 - y \rangle)$. Let $X \rightarrow \mathbb{A}^1$ be the projection map corresponding to the map $k[t] \rightarrow k[x, y]$ sending $t$ to $x^2 = y$. Is $X \rightarrow \mathbb{A}^1$ flat?